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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Find an equation of the form ρ = ƒ(θ, φ) in spherical coordinates for the following surfaces.z2 = 3(x2 + y2)
Find an equation of the form ρ = ƒ(θ, φ) in spherical coordinates for the following surfaces.x = z2
Find an equation of the form ρ = ƒ(θ, φ) in spherical coordinates for the following surfaces.z = x2 + y2
Find an equation of the form ρ = ƒ(θ, φ) in spherical coordinates for the following surfaces.x2 − y2 = 4
Which of (a)–(c) is the equation of the cylinder of radius R in spherical coordinates? Refer to Figure 17. (a) Rp = sin o (b) p sin = R Z R (c) p = Rsino
Find an equation of the form ρ = ƒ(θ, φ) in spherical coordinates for the following surfaces.xy = z
Let P1 = (1, − √3, 5) and P2 = (−1,√3, 5) in rectangular coordinates. In which quadrants do the projections of P1 and P2 onto the xy-plane lie? Find the polar angle θ of each point.
Find the spherical angles (θ, ϕ) for Helsinki, Finland (60.1° N, 25.0° E), and São Paulo, Brazil (23.52° S, 46.52° W).
Find the longitude and latitude for the points on the globe with angular coordinates (θ, φ) = (π/8, 7π/12) and (4, 2).
Consider a rectangular coordinate system with its origin at the center of the earth, z-axis through the North Pole, and x-axis through the prime meridian. Find the rectangular coordinates of Sydney, Australia (34° S, 151° E), and Bogot´a, Colombia (4°32 N, 74° 15 W). A minute is 1/60°.
Find the equation in rectangular coordinates of the quadric surface consisting of the two cones ϕ = π/4 and ϕ = 3π/4.
Find an equation of the form z = f (r, θ) in cylindrical coordinates for z2 = x2 − y2.
Show that ρ = 2 cos ϕ is the equation of a sphere with its center on the z-axis. Find its radius and center.
An apple modeled by taking all the points in and on a sphere of radius 2 inches is cored with a vertical cylinder of radius 1 inch. Use inequalities in cylindrical coordinates to describe the set of all points that remain in the apple once the core is removed.
Repeat Exercise 81 using inequalities in spherical coordinates.Data From Exercise 81An apple modeled by taking all the points in and on a sphere of radius 2 inches is cored with a vertical cylinder of radius 1 inch. Use inequalities in cylindrical coordinates to describe the set of all points that
Explain the following statement: If the equation of a surface in cylindrical or spherical coordinates does not involve the coordinate θ, then the surface is rotationally symmetric with respect to the z-axis.
A great circle on a sphere S with center O and radius R is a circle obtained by intersecting S with a plane that passes through O (Figure 19). If P and Q are not antipodal (on opposite sides), there is a unique great circle through P and Q on S (intersect S with the plane through O, P, and Q). The
Find equations r = g(θ, z) (cylindrical) and ρ = ƒ(θ, ф) (spherical) for the hyperboloid x2 + y2 = z2 + 1 (Figure 18). Do there exist points on the hyperboloid with ф = 0 or π? Which values of ф occur for points on the hyperboloid? Z
Plot the surface ρ = 1 − cos ф. Then plot the trace of S in the xz-plane and explain why S is obtained by rotating this trace.
Show that the geodesic distance from Q = (a, b, c) to the North Pole P = (0, 0, R) is equal to Rcos- -1 IR .
Show that the central angle ψ between points P and Q on a sphere (of any radius) with angular coordinates (θ, ϕ') and (θ, θ') is equal to = cos (sin sin d' cos(0-8) + cos o cos ')
The coordinates of Los Angeles are 34° N and 118° W. Find the geodesic distance from the North Pole to Los Angeles, assuming that the earth is a sphere of radius R = 6370 km.
Use Exercise 89 to find the geodesic distance between Los Angeles (34° N, 118° W) and Bombay (19° N, 72.8° E).Data From Exercise 89Show that the central angle ψ between points P and Q on a sphere (of any radius) with angular coordinates (θ, ϕ') and (θ, θ') is equal to = cos(sino sin '
Let a, b, c be nonzero vectors. Assume that b and c are not parallel, and set v = ax (bx c), (a) Prove that: (i) v lies in the plane spanned by b and c. (ii) v is orthogonal to a. w = (a c)b (a . b)c . (b) Prove that w also satisfies (i) and (ii). Conclude that v and w are parallel. (e) Show
The torque about the origin O due to a force F acting on an object with position vector r is the vector quantity τ = r × F. If several forces Fj act at positions rj, then the net torque (units: N-m or lb-ft) is the sumLet τ be the net torque about O acting on the robotic arm of Figure 22, here
The torque about the origin O due to a force F acting on an object with position vector r is the vector quantity τ = r × F. If several forces Fj act at positions rj, then the net torque (units: N-m or lb-ft) is the sumCalculate the net torque about O at P, assuming that a 30-kg mass is attached
The torque about the origin O due to a force F acting on an object with position vector r is the vector quantity τ = r × F. If several forces Fj act at positions rj, then the net torque (units: N-m or lb-ft) is the sumCalculate the torque τ about O acting at the point P on the mechanical arm in
Solve the equation (1, 1, 1) × X = (1, −1, 0), where X = (x, y, z). There are infinitely many solutions.
Use the result of Exercise 61 to determine whether the points P, Q, and R are collinear, and if not, find a vector perpendicular to the plane containing them.Data From Exercise 61Show that three points P, Q, R are collinear (lie on a line) if and only if (a) P = (2, 1,0), (b) P = (2,
Calculate the 2 × 2 determinant. 1 14 2 3
Calculate the 2 × 2 determinant. NIM 3 -5 21
Calculate the 2 × 2 determinant. 1
Calculate the 2 × 2 determinant. 9 25 5 14
Calculate the 3 × 3 determinant. 4 2 -3 0 1| 0 1
Calculate the 3 × 3 determinant. 1 -2 1 0 0 3 3 -1
Calculate the 3 × 3 determinant. 1 2 -3 2 4 -4 3 6 لنا
Calculate the 3 × 3 determinant. |1 0 0 이 0 0- 1 - 1 0
Calculate v × w. v = (1, 2, 1), w = (3, 1, 1)
Calculate v × w. v = (2,0,0), w = (-1,0,1)
Calculate v × w. V =(3, 1, 2), w=(4,-6,3)
Calculate v × w. v= (1, 1,0), w = (0, 1, 1)
Calculate v × w. V= v = (2.4, -1.25,3), w = (-7.68, 4,-9.6)
Use the relations in Eqs. (5) and (6) to calculate the cross product.(i + j) × k ixj=k, jxi = -k, jxk=i, kx j = -i, kxi=j ixk=-j
Use the relations in Eqs. (5) and (6) to calculate the cross product.( j − k) × ( j + k) ixj=k, jxi = -k, jxk=i, kx j = -i, kxi=j ixk=-j
Use the relations in Eqs. (5) and (6) to calculate the cross product.(i − 3j + 2k) × ( j − k) ixj=k, jxi = -k, jx k=i, kx j = -i, kxi=j ixk=-j
Use the relations in Eqs. (5) and (6) to calculate the cross product.(2i − 3j + 4k) × (i + j − 7k) ixj=k, jxi = -k, jxk=i, kx j = -i, kxi=j ixk=-j
Calculate the cross product assuming thatv × u ux v = (1, 1, 0), uxw = (0, 3, 1), vxw= (2,-1, 1)
Calculate the cross product assuming thatv × (u + v) ux v = (1, 1,0), uxw = (0, 3, 1), vX W= (2,-1, 1)
Calculate the cross product assuming thatw × (u + v) ux v = (1, 1, 0), uxw = (0, 3, 1), vxw= (2,-1, 1)
Calculate the cross product assuming that(3u + 4w) × w ux v= (1, 1,0), uxw = (0, 3, 1), vX W= (2,-1, 1)
Calculate the cross product assuming that(u − 2v) × (u + 2v) ux v = (1, 1, 0), uxw = (0, 3, 1), vxw= (2,-1, 1)
Calculate the cross product assuming that ux v = (1, 1, 0), uxw = (0, 3, 1), vxw= (2,-1, 1)
Find v × w, where v and w are vectors of length 3 in the xz-plane, oriented as in Figure 16, and θ = π/6. X 3 W 3 0
Refer to Figure 17.Which of u and −u is equal to v × w? n -u n- W
Let v = (a, b, c). Calculate v × i, v × j, and v × k.
Refer to Figure 17.Which of the following form a right-handed system? n -u n- W
Let v = 3, 0, 0 and w = 0, 1, −1. Determine u = v × w using the geometric properties of the cross product rather than the formula.
What are the possible angles θ between two unit vectors e and f if ||e × f||= 1/2?
Show that if v and w lie in the yz-plane, then v × w is a multiple of i.
Find the two unit vectors orthogonal to both a = (3, 1, 1) and b = (−1, 2, 1).
Let e and e' be unit vectors in R3 such that e ⊥ e'. Use the geometric properties of the cross product to compute e × (e' × e).
Determine the magnitude of each Coriolis force on a 1.5-kg parcel of air with wind v.(a) v is 25 m/s toward the east at the equator(b) v is 25 m/s toward the east at 45°N(c) v is 35 m/s toward the south at 30°N(d) v is 35 m/s toward the south at 60°N
Determine the magnitude of each Coriolis force on a 2.3-kg parcel of air with wind v.(a) v is a 20 m/s toward the west at the equator(b) v is a 20 m/s toward the west at 60°N(c) v is a 40 m/s toward the south at the equator(d) v is a 40 m/s toward the south at 45°S
Assume an electron moves with velocity v in the plane and B is a uniform magnetic field pointing directly out of the page. Which of the two vectors, F1 or F2, in Figure 18 represents the force on the electron? Remember that q is negative. I F₁ F₂ B. I
A force F (in newtons) on an electron moving at velocity v meters per second in a uniform magnetic field B (in teslas) is given by F = q(v × B), where q = −1.6 × 10−19 coulombs is the charge on the electron.Calculate the force F on an electron moving with velocity 105 m/s in the direction i
Find the volume of the parallelepiped spanned by u, v, and w in Figure 19. u = (1, 0, 4) X N w = (-4, 2, 6) v = (1,3,1) y
Calculate the scalar triple product u · (v × w), where u = (1, 1, 0), v = (3, −2, 2), and w = (4, −1, 2).
Verify identity (12) for vectors v = (3, −2, 2) and w = (4, −1, 2).
Find the area of the parallelogram spanned by v and w in Figure 19. u= (1, 0, 4) X Z -w=(-4, 2, 6) v= (1, 3, 1) y
Calculate the volume of the parallelepiped spanned by u = (2, 2, 1), V = (1, 0,3), W = (0,-4, 0)
Sketch and compute the volume of the parallelepiped spanned by u = (1, 0, 0), V v = (0,2,0), w = (1,1,2)
Sketch the parallelogram spanned by u = (1, 1, 1) and v = (0, 0, 4), and compute its area.
Calculate the area of the parallelogram spanned by u = (1, 0, 3) and v = (2, 1, 1).
Find the area of the parallelogram determined by the vectors (a, 0, 0) and (0, b, c).
Sketch the triangle with vertices at the origin O, P = (3, 3, 0), and Q = (0, 3, 3), and compute its area using cross products.
Use the cross product to find the area of the triangle with vertices P = (1, 1, 5), Q = (3, 4, 3), and R = (1, 5, 7).
Use the cross product to find the area of the triangle in the xy-plane defined by (1, 2), (3, 4), and (−2, 2).
Use the cross product to find the area of the quadrilateral in the xy-plane defined by (0, 0), (1, −1), (3, 1), and (2, 4).
Check that the four points P(2, 4, 4), Q(3, 1, 6), R(2, 8, 0), and S (7, 2, 1) all lie in a plane. Then use vectors to find the area of the quadrilateral they define.
Use the cross product to find the area of the triangle with vertices (a, 0, 0), (0, b, 0), and (0, 0, c).
Verify the identity using the formula for the cross product.v × w = −w × v
Verify the identity using the formula for the cross product. (lv) x W = 1(vx w) (a scalar)
Verify the identity using the formula for the cross product. (u + v) x W = uxW+Vxw
Prove i × j = k and k × j = −i by each of the following methods:(a) Using the definition of cross product as a determinant(b) Using the geometric description of the cross product in Theorem 1 THEOREM 1 Geometric Description of the Cross Product Given two nonzero non- parallel vectors v and w
Using standard basis vectors, find an example demonstrating that the cross product is not associative.
The components of the cross product have a geometric interpretation. Show that the absolute value of the k-component of v × w is equal to the area of the parallelogram spanned by the projections v0 and w0 onto the xy-plane (Figure 20). X V I IW Wo
Assume that v and w lie in the first quadrant in R2 as in Figure 24. Use geometry to prove that the area of the parallelogram is equal to det W
Show that if a, b are nonzero vectors such that a ⊥ b, then the set of all solutions of Eq. (15) is a line with a as direction vector.Show that if X is orthogonal to b and is not a multiple of a, then a × X is a multiple of b. axx=b
Suppose that u, v, w are nonzero andShow that u, v, and w are either mutually parallel or mutually perpendicular. Use Exercise 75.Data From Exercise 75Show that if u, v, and w are nonzero vectors and (u × v) × w = 0, then either (i) u and v are parallel, or (ii) w is orthogonal to u and v. (u xv)
Show that if a, b are nonzero vectors such that a ⊥ b, then there exists a vector X such thata × X = b
Use Eq. (12) to prove the Cauchy-Schwarz inequality:Show that equality holds if and only if w is a multiple of v or at least one of v and w is zero. z(M·A) - z||M||₂||A|| = ₂||MX A||
Show that if u, v, and w are nonzero vectors and (u × v) × w = 0, then either (i) u and v are parallel, or (ii) w is orthogonal to u and v.
Use the diagonal rule to calculate 2 0 4 31 1-7. 5 3
Show that 3 × 3 determinants can be computed using the diagonal rule: Repeat the first two columns of the matrix and form the products of the numbers along the six diagonals indicated. Then add the products for the diagonals that slant from left to right and subtract the products for the diagonals
Let X = (x, y, z). Show that i × X = v has a solution if and only if v is contained in the yz-plane (the i-component is zero).
Explain geometrically why (1, 1, 1) × X = (1, 0, 0) has no solution, where X = (x, y, z).
Consider the tetrahedron spanned by vectors a, b, and c as in Figure 25(A). Let A, B, C be the faces containing the origin O, and let D be the fourth face opposite O. For each face F, let vF be the vector that is perpendicular to the face, pointing outside the tetrahedron, of magnitude equal to
Calculate the area of the circle r = 4 sin θ as an integral in polar coordinates (see Figure 4). Be careful to choose the correct limits of integration. RIN 2 24 5x 12 RIM KA (A) The polar integral computes the area swept out by a radial segment. 2 (B) The ordinary integral in rectangular
Find parametric equations for the circle of radius 2 with center (1, 1). Use the equations to find the points of intersection of the circle with the x- and y-axes.
Find a parametrization c(t) of the line y = 5 − 2x such that c(0) = (2, 1).

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